4.1 Archimedes' Principle and Buoyancy

Archimedes' Principle and Buoyancy

Archimedes' principle is the foundation of buoyancy in fluid mechanics. It states that the upward force on an object in a fluid equals the weight of the fluid that object displaces. This single idea explains why massive steel ships float, why submarines can control their depth, and how engineers design everything from offshore platforms to hydraulic systems.

Whether an object floats, sinks, or hovers in place depends on the balance between its weight and the buoyant force acting on it. The sections below cover how to calculate buoyant force, determine floating vs. sinking conditions, and analyze the equilibrium and stability of floating objects.

Archimedes' Principle in Fluid Mechanics

Archimedes' principle says that the buoyant force $F_b$ acting on an object immersed in a fluid equals the weight of the fluid displaced by that object. In equation form:

$F_b = \rho_{fluid} \, V_{displaced} \, g$

where $\rho_{fluid}$ is the fluid's density, $V_{displaced}$ is the volume of fluid pushed aside by the object, and $g$ is gravitational acceleration.

A few things to keep in mind:

• The buoyant force always acts upward, through the centroid of the displaced fluid volume (more on this below).
• The principle applies to both liquids and gases, since both are fluids. A helium balloon floats in air for the same reason a cork floats in water: the displaced fluid weighs more than the object. For most solid objects in air, though, the buoyant force is negligibly small compared to their weight.
• This principle is used constantly in engineering: predicting ship draft, sizing hydraulic lifts, and analyzing the stability of floating structures like buoys and offshore platforms.

Calculation of Buoyant Force

The calculation differs slightly depending on whether the object is fully or partially submerged.

Fully submerged object:

1. The displaced volume equals the object's entire volume $V_{object}$.
2. Buoyant force: $F_b = \rho_{fluid} \, V_{object} \, g$

Partially submerged object:

1. Only the portion below the fluid surface displaces fluid.
2. Buoyant force: $F_b = \rho_{fluid} \, V_{submerged} \, g$

In both cases, the buoyant force acts through the center of buoyancy, which is the centroid of the displaced fluid volume. This is not necessarily the same as the object's center of gravity. The relative positions of these two points determine stability, which is covered in the equilibrium section below.

Conditions for Floating vs. Sinking

What determines whether an object floats or sinks comes down to comparing the object's density to the fluid's density.

• Floating ($\rho_{object} < \rho_{fluid}$): The object displaces enough fluid before it's fully submerged to support its own weight. It settles at a depth where $F_b = W_{object}$. Examples: wood in water ($\rho_{wood} \approx 500{-}700 \, \text{kg/m}^3$), oil floating on water.
• Sinking ($\rho_{object} > \rho_{fluid}$): Even fully submerged, the object can't displace enough fluid to match its own weight, so there's a net downward force. Examples: a stone in water, solid steel in water ($\rho_{steel} \approx 7850 \, \text{kg/m}^3$).
• Neutral buoyancy ($\rho_{object} = \rho_{fluid}$): The object's weight exactly equals the buoyant force when fully submerged. It will remain suspended at whatever depth it's placed, with no tendency to rise or sink. Fish achieve this by adjusting their swim bladders, and submarines do it by flooding or emptying ballast tanks.

Equilibrium of Floating Objects

For a floating object at rest, the buoyant force and the object's weight are in balance:

$F_b = W_{object}$

Expanding this:

$\rho_{fluid} \, V_{displaced} \, g = m_{object} \, g$

The $g$ cancels, giving a useful relationship for finding how much of the object sits below the surface:

$V_{displaced} = \frac{m_{object}}{\rho_{fluid}}$

This tells you the submerged volume directly. For example, an iceberg with density $\approx 917 \, \text{kg/m}^3$ floating in seawater ($\approx 1025 \, \text{kg/m}^3$) has about $917/1025 \approx 89.5\%$ of its volume submerged. That's why only a small fraction is visible above the waterline.

Stability of floating objects:

Stability depends on the relative positions of three points:

• Center of gravity (G): where the object's weight acts.
• Center of buoyancy (B): the centroid of the displaced fluid volume.
• Metacenter (M): the point where the line of action of the buoyant force (when the object is slightly tilted) intersects the original vertical line through B.

The stability rule is straightforward:

• If M is above G, the object is stable. A small tilt creates a restoring moment that pushes it back upright.
• If M is below G, the object is unstable. A small tilt creates an overturning moment, and the object capsizes.
• If M coincides with G, the object is in neutral equilibrium.

Ship designers pay close attention to the metacentric height $GM$ (the distance from G to M). A larger $GM$ means greater stability, which is why cargo ships are loaded so that heavy items sit low, keeping G well below M.